A chiral aperiodic monotile

TL;DR

The paper addresses whether a single tile can tile the plane aperiodically under translations and rotations, and whether weakly or strictly chiral aperiodic monotiles exist. It shows $Tile(1,1)$ is a weakly chiral aperiodic monotile, and by edge-modifying it to obtain Spectres, constructs strictly chiral aperiodic monotiles that admit only homochiral tilings via a hierarchical substitution framework. A combinatorial bridge to hats and turtles, plus computer-assisted patch enumeration, underpins the proofs and leads to a substitution system on nine marked hexagons (via a Mystic) with an irrational growth factor $4+ obreak\ obreak ( obreak √{15})$, ensuring non-periodicity. These results extend the landscape of einstein tilings by demonstrating strictly chiral aperiodic monotiles closely related to the hat-turtle continuum and provide a practical pathway to constructing and analyzing such tilings.

Abstract

The recently discovered "hat" aperiodic monotile mixes unreflected and reflected tiles in every tiling it admits, leaving open the question of whether a single shape can tile aperiodically using translations and rotations alone. We show that a close relative of the hat -- the equilateral member of the continuum to which it belongs -- is a weakly chiral aperiodic monotile: it admits only non-periodic tilings if we forbid reflections by fiat. Furthermore, by modifying this polygon's edges we obtain a family of shapes called Spectres that are strictly chiral aperiodic monotiles: they admit only chiral non-periodic tilings based on a hierarchical substitution system.

Figures (12)

• Figure 1.1: The $14$-sided polygon $\mathrm{Tile}(1,1)$, shown on the left, is a weakly chiral aperiodic monotile: if by fiat we forbid tilings that mix unreflected and reflected tiles, then it admits only non-periodic tilings. By modifying its edges, as shown in the centre and right for example, we obtain strictly chiral aperiodic monotiles called "Spectres" that admit only non-periodic tilings even when reflections are permitted.
• Figure 1.2: A patch from a non-periodic tiling by Spectres. On the left, tiles are drawn as copies of $\mathrm{Tile}(1,1)$, and thickened outlines show the hierarchy of supertiles to which these tiles will be shown to belong. On the right, tile boundaries are modified in a manner similar to Figure \ref{['fig:polygonspectre']} (right). Tile colours will be explained later in the paper.
• Figure 2.1: Two substitution rules that can be iterated to tile the plane with Spectres. The rules are based on a single Spectre and a two-Spectre compound called a Mystic. The first rule (left) replaces the Spectre by a cluster containing a Mystic and seven Spectres, all reflected; the second (right) replaces a Mystic by a cluster containing a Mystic and six Spectres. In Section \ref{['sec:hexagons']} we show that every tiling by Spectres can be composed into non-overlapping congruent copies of these clusters.
• Figure 2.2: Five levels of supertiles created by applying the substitution rules given in Figure \ref{['fig:thespectretiles']}, starting with a single Spectre. The Mystic and its supertiles are shaded in progressive levels of green. Each step uses Spectres of the opposite handedness of the one before it.
• Figure 3.1: Tilings by $\mathrm{Tile}(1,1)$ are combinatorially equivalent to tilings by hats and turtles, through a change to the lengths of the "even" and "odd" tile edges, oriented at even and odd multiples of $30^\circ$ to the horizontal (top right). All three patches are drawn so that short edges have unit length.

Theorems & Definitions (11)

• Lemma 2.1
• proof
• Theorem 2.2
• Lemma 2.3
• proof
• Theorem 3.1
• proof
• Theorem 4.1
• proof : Proof of Theorem \ref{['thm:clusters89']}
• Theorem 5.1